The next chapter begins our discussion of Kurt Gödel's famous incompleteness theorems. Rather than discussing the strength of our deductive system as we have done in the last two chapters, we will now discuss the strength of sets of axioms. In particular, we will look at the question of how complicated a set of axioms must be in order to prove all of the true statements about the standard structure $$\mathfrak{N}$$.
In Chapter 4 we will introduce the idea of coding up the statements of $$\mathcal{L}_{NT}$$ as terms and will show that a certain set of nonlogical axioms is strong enough to prove some basic facts about the numbers coding up those statements. Then, in Chapters 5 and 6, we will bring those facts together to show that the expressive power we have gained has allowed us to express truth that are unprovable from our set of axioms.